Below is shown the graph of f, a secant and the two tangent corresponding to the two solutions found. The secant and the two tangents are parallel since their slopes are equal according to the mean value theorem.

Problem 2:Use the mean value theorem to prove that for any two real numbers a and b,

| cos a - cos b| <= | a - b|

Solution to Problem 2

Function cos x is continuous and differentiable for all real numbers. Use the mean value theorem, using 2 real numbers a and b to write